\section{Introduction}
\label{sec:introduction}

The Local Extreme Learning Machine (locELM) method proposed by Dong and Li (2020) combines domain decomposition with shallow neural networks whose hidden-layer parameters are fixed at random values while only the output layer is trained. The reference study reported both high accuracy and competitive runtime on canonical Helmholtz benchmark problems. The present reproduction effort implements the method as a Python package and evaluates how closely the published results can be matched on commodity hardware.

This document distills the key information captured across the project's Markdown documentation. We summarize the implemented software architecture, reproduce the experimental results that were obtained, and discuss the remaining gaps relative to the paper. The goal is to provide a coherent narrative that can guide further debugging, optimization, and validation work.
